What is the integral of ((ln(x))^2+1)/(ln(x)) ?

The integral of ((ln(x))^2+1)/(ln(x)) is xln(x)-x+\Ei(ln(x))+C

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integral of ((ln(x))^2+1)/(ln(x))

\int \frac{( ln ( x ) ) ^{2} + 1}{ln ( x )} d x

x ln (x) - x + Ei (ln (x)) + C

Solution steps

\int \frac{( ln ( x ) ) ^{2} + 1}{ln ( x )} d x

Expand

\frac{( ln ( x ) ) ^{2} + 1}{ln ( x )} : ln (x) + \frac{1}{ln ( x )}

= \int ln (x) + \frac{1}{ln ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int ln (x) d x + \int \frac{1}{ln ( x )} d x

\int ln (x) d x = x ln (x) - x

\int \frac{1}{ln ( x )} d x = Ei (ln (x))

= x ln (x) - x + Ei (ln (x))

Add a constant to the solution

= x ln (x) - x + Ei (ln (x)) + C

y^{^{'}} = \frac{e ^{x - y}}{e ^{x} + 1}

\frac{d ^{2} y}{d x ^{2}} - 5 \frac{d y}{d x} + k y = 0

y^{^{'}} + 3 x e^{y} = 0

n = 1 \sum \infty \frac{3 ^{n}}{n 2 ^{n}}

\frac{\partial}{\partial y} (arctan (\frac{y}{x}))

What is the integral of ((ln(x))^2+1)/(ln(x)) ?
The integral of ((ln(x))^2+1)/(ln(x)) is xln(x)-x+\Ei(ln(x))+C